A Codimension Two Approach to the $\mathbb{S}^1$-Stability Conjecture
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912459109957632 |
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| author | Rosenberg, Steven Xu, Jie |
| author_facet | Rosenberg, Steven Xu, Jie |
| contents | J. Rosenberg's $\mathbb{S}^1$-stability conjecture states that a closed oriented manifold $X$ admits a positive scalar curvature metric iff $X\times \mathbb{S}^1$ admits a positive scalar curvature metric $h$. As pointed out by J. Rosenberg and others, there are known counterexamples in dimension four. We prove this conjecture whenever $h$ satisfies a geometric bound which measures the discrepancy between $\partial_θ\in T\mathbb{S}^1$ and the normal vector field to $X\times \{P\}$, for a fixed $P\in \mathbb{S}^1.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_12479 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Codimension Two Approach to the $\mathbb{S}^1$-Stability Conjecture Rosenberg, Steven Xu, Jie Differential Geometry 53C21 J. Rosenberg's $\mathbb{S}^1$-stability conjecture states that a closed oriented manifold $X$ admits a positive scalar curvature metric iff $X\times \mathbb{S}^1$ admits a positive scalar curvature metric $h$. As pointed out by J. Rosenberg and others, there are known counterexamples in dimension four. We prove this conjecture whenever $h$ satisfies a geometric bound which measures the discrepancy between $\partial_θ\in T\mathbb{S}^1$ and the normal vector field to $X\times \{P\}$, for a fixed $P\in \mathbb{S}^1.$ |
| title | A Codimension Two Approach to the $\mathbb{S}^1$-Stability Conjecture |
| topic | Differential Geometry 53C21 |
| url | https://arxiv.org/abs/2412.12479 |